Like everyone else, I suppose, I have a need to express my creativity. To facilitate this I am coding a small programming language to create graphical art. Although, for me, Mathematica is -the- language to do that I want to have a language that kids of age 8 to 88 ( TinTin or Donald Duck readers ) could learn. By using it geometric transformations like rotations and translations are learned in a natural way.
The graphic like the one below, including the rotation, is an example what the language could create with a few lines, a Mathematica Kernel is required but will run in the background only.
In the Open University course Number Theory and Mathematical Logic the additional exercises sections of the workbooks are complemented with several 'challenge exercises'. This is one of them.
Let $n$ be an odd positive integer.
Prove that there are $\tau(n)$ ways of writing $n$ as a sum of consecutive positive integers.
For example, if $n=9$, $\tau(9)=3$ because $9$ has three divisors $1,3,9$ and the three sums are: $9$, $4+5$ and $2+3+4$.
In the book 'Problem Solving and Number Theory' I read
The law of quadratic reciprocity was discovered for the first time, in a complex form, by L. Euler who published it in his paper entitled “Novae demonstrationes circa divisores numerorum formae $xx + nyy$.
Was there no notation for exponents in the time of the great Euler? I did not know how to formulate a query for Google, so I asked a question at Math/StackExchange.
I found out that Descartes introduced the notation for $x^2$. Descartes is famous for his quote "Cogito ergo sum", "Je pense donc je suis" or in plain English "I think, therefore I am". Descartes was my favourite mathematician in secondary school because he invented analytic geometry, one of the milestones in the development of mathematics. Before Descartes geometry was done strictly in the Euclidean way, by compass and ruler. Thanks to Descartes' quadrant and coordinates, geometric shapes like lines and circles could be respresented by algebraic equations. They become objects to do calculations with.
So far about Descartes.
Also, thanks to kind repliers I discovered the bookset "A History of Mathematical Notations, vols 1 and 2., by Florian Cajori". A real gem if you like the history of mathematics. Because the copyright expired one is allowed to freely download the original version. See the comments below the original question at StackExchange for a link to the full version of the book.
The places 'to be' for mathematicians are -not- Twitter and facebook, although they are both great places to socialize, chit-chat you know what I mean, they basically steal your valuable time.
Numerologist is a four-letter word among mathematicians ( I can imagine ). Is numerology the same as number theory? Not really, the difference between numerology and number theory, as a branch of mathematics, can best be explained using an example.
Both numerologists and mathematicians study identities like $$2^2 + 3^2 + 5^2 + 7^2 + 11^2 + 13^2 + 17^2 = 666.$$ A mathematician would notice that this is the sum of the first seven primes and rewrite the sum as $$\sum_{k=1}^{7} p_k = 666.$$ The right-hand side sum is a number with equal digits, so the mathematician might look at other sums of consecutive primes and verify if these sums have similar patterns. Basically a mathematician is interested in anything that might lead to the formulation of a theorem, a proposition to prove mathematically.
A numerologist would immediately notice that $666$ is the number of "The Beast", the representation of evil in the Christian belief system. Then, a numerologist might consider prime numbers divine, since they are the building blocks of all integers, and might try to formulate some law of good and bad represented in the sum-formula. Numerologists also believe that future events can be predicted so they will be extra alert to that.
I believe that it was until the Middle Ages that there was no real distinction between the profession of numerologist and mathematician. Newton has been called the last Alchemist, perhaps he was the last mathematician / numerologist as well. ( History, I am afraid, is not my strongest point. )
This is as yet the last post in the LQR series. If you are only interested in -solving- ( quadratic ) congruence equations then this is the way to do it in Mathematica:
As a general Diophantic equation: In[1]:= Reduce[x^2==123456+1299709 k,{x,k},Integers]
M381 unit 6 is about the Law of Quadratic Reciprocity. An application of the LQR is solving quadratic modular equations like: $$x^2 \equiv 499 \ \text{mod(617)}.$$ M381 contains a method that can fast determine if that equation is solvable. It does not contain however a fast algorithm for finding the actual solutions. One such algorithm is the Shanks-Tonelli algorithm and can be found on Planet Math which as often gives a much clearer presentation than the messy Wikipedia-entry.
Let $p$ and $q$ be distinct odd primes. Then $$\displaystyle \left({\frac p q}\right) \left({\frac q p}\right) = \left({-1}\right)^{\frac {\left({p-1}\right) \left({q-1}\right)} 4}$$ where $\displaystyle \left({\frac p q}\right)$ and $\displaystyle \left({\frac q p}\right)$ are defined as the Legendre Symbol $\displaystyle \left({\frac{a}{p}}\right) := a^{\frac{(p-1)}{2}} \pmod p$.
Gauss considered his work on the Quadratic Reciprocity Law among his major achievements. I don't 'get that', not now anyway, that's a call for more study on the topic.
Now and then, when I browse through papers, or otherwise, I find an interesting mathematical paper... ( that I can actually read ). Actually, I was browsing through a book called Reciprocity Laws, from Euler to Eisenstein by Franz Lemmermeyer, it contains more than 100 proofs of the Quadratic Reciprocity Law. I hoped to find a proof I could appreciate by it's beauty. Although most proofs are based on Gauss's Lemma ( as the proof in M381 ) but there are proofs in other realms of mathematics like Group Theory. Group Theory -as we know it today- did not exist in Gauss's time. That's why I am going to spend some time studying the following paper 'Quadratic reciprocity in a finite group.'
A wise man rode into a desert village one evening as the sun was setting. Dismounting from his camel, he asked one of the villagers for a drink of water.‘Of course,’ said the villager and gave him a cup of water. The traveller drank the whole cupful. ‘Thank you,’ he said. ‘Can I help you at all before I travel on?’‘Yes,’ said the young man. ‘We have a dispute in our family. I am the youngest of three brothers. Our father died recently, God rest his soul, and all he possessed was a small herd of camels. Seventeen, to be exact. He decreed in his will that one half of the herd was to go to my oldest brother, one third to the middle brother and one ninth to me. But how can we divide a herd of 17? We do not want to chop up any camels, they are worth far more alive.’ ‘Take me to your house,’ said the sage. When he entered the house he saw the other two brothers and the man’s widow sitting around the fire arguing. The youngest brother interrupted them and introduced the traveller.
‘Wait,’ said the wise man, ‘I think I can help you. Here, I give you my camel as a gift. Now you have 18 camels. One half goes to the eldest, that’s nine camels. One third goes to the middle son, that’s six camels. And one ninth goes to my friend here, the youngest son. That’s two.’ ‘That’s only 17 altogether,’ said the youngest son. ‘Yes. By a happy coincidence, the camel left over is the one I gave to you. If you could possibly give it back to me, I will continue on my journey.’ And he did.
Answering this question requires a definition of God, which we don't have of course. We could propose a temporarily definition like 'God is the entity that created life on planet Earth, a modest definition because it leaves the question of who created the universe unanswered.
Although still debated, Gödel's incompleteness theorems imply that we won't be able to build computers with a conscious mind capable of creating new mathematics or writing a program that solves all open problems in mathematics. Besides cloning and modifying what we have ( DNA ) we won't be able to create a 'machine' smarter than ourselves.
Our own existence is a paradox we don't understand. History proves that in cases like that humans are creative in inventing a God capable of fixing any problem. Despite centuries of scientific advances we still need a God to explain our existence.
Was it Cairo or New York, - Occupy Wall Street - where it started? It sure is spreading. I want to express my solidarity with the people out on the streets today in almost 1600 cities world-wide: Occupy Together. Since the revolution started in Egypt I watched and compared the news from BBC, CNN, Infowars, Al Jazeera, RT ( Russia ), and Press TV ( Iran ). Now Press TV has been taken off-air in the UK. Draw your own conclusions. Here is what Infowars ( US Libertarian ) had to say about it: Press TV taken off air in UK – war for freedom of speech now on. Now that the alternative media are flourishing, it turns out that we haven't had a free press for decades, Press TV made that painfully clear. During the last world war, scientists created the atom bomb, period. 'Because they were afraid the Germans would be first.' We have to assure that they don't have an easy excuse like that next time, a free Internet ensures that.
Did M381 exam. All the questions were doable, easy as a matter of fact. Honestly. I am sure I have full marks for the first question I made. But when I was done and looked at the clock it was already past three o'clock. "Should I have been able to do that question in 10 minutes?", I thought. Maybe. I have a painful ear-infection at the moment, and I was drugged of course, prescription painkillers, doctor's order. And antibiotics, of course. I felt like walking on the moon, in an astronaut's suit. Whatever the outcome may be, I can set it right. More later on the exam, when I feel better.
A lot has been written about real paper books versus ebooks. Both have their distinct advantages and disadvantages. I have to admit that I read mostly ebooks. They are cheaper and easier to get, store and carry. Mathematics ebooks often have the PDF format. So you need a reader that can handle PDFs. If you use your PC or laptop than a PDF reader is all you need. Most people know Adobe Reader but there are much better programs than Adobe around. With free readers, just like Adobe. My PDF reader of choice is Foxit Reader 5. What I particularly like about Foxit is that it is lightweight, i.e. loads and acts fast. Foxit uses tabbed reading, like internet browsers. If you were in the middle of five books, close Foxit, the program nicely remembers which books you were reading and on what page you left. Most of all, I like the feature that I can highlight what I read. It is almost as if I was reading in a paper book.
Only recently lightspeed as the ultimate limit of speed got challenged. It seems they have been challenging the Turing Machine for a while too.
If you start with studying mathematics you are only three or four centuries behind on contemporary mathematics. That's quite a lot of catching up to do. - Some fields started their development in the previous century though. Like mathematical logic, a field I have been studying this year, and have written about so now and then in this blog. Part of mathematical logic is the theory of computation which showed us what can be computed and what can't. That what can be computed is what can be computed on a Turing Machine, period. - That idea is challenged however in the theory of hypercomputation. A new field in mathematics which is trying to go beyond the limits of the Turing Machine.
A book with an overview of the theory is the following.
The Occupy Wall Street movement is spreading. It reminds me of Conway's Life game. It could spread enormously and still die out soon, or it could cause permanent change without real massive demonstrations. Nobody knows, nobody can predict this. Still, I think it will be very interesting to find mathematical patterns in global, Internet connected, demonstrations like this. I am sure mathematicians of government agencies are working on it.
If a government does not educate even one generation it is lost. It is in their own interest, it is in the interest of the ruling elite to give all citizens a good education. I think that rule fits all political systems. In the US however they use education to enslave people for the rest of their life to the bankers. Paying back student loans turns out to be very difficult. On top of that teachers get fired or are underpaid. - That does not seem right to me.
Normally you calculate n factorial with Factorial[n] or short n!. Mathematica handles the details of the function for you and prints the result.
In the course M381 you have to prove that functions like factorial are a primitive recursive function. This basically means that the function can be defined only in terms of itself, add one, or set to zero. A primitive recursive definition of factorial would look as follows in Mathematica.
Mathematicians have a tendency to regard texts which are not written using 'protocol' as irrelevant. Long ago I wrote a note to a mathematician and his reply was that I should formulate my thoughts in 'standard mathematics'. I did my very best 'to make myself clear'. It was not enough. The thing is mathematicians lose their authority when they leave familiar territory. Well, at least I received a reply. ( Although that was all he did. And I haven't given up on the problem I was working on... )
The blog received a comment, containing what I, for the moment, call 'out-of-the-box' thinking. Non 'standard mathematics' at least. I have moved the comment to this post in an attempt to share this with as much as possible readers. I will reply. But later, I have to let it work on me first.
Regarding our active internal analog math...
I'm a civil/environmental engineer by education but I've been working off and on on a theory which takes the tact that all abstract math symbols and expressions are secondary and arise from a handful of internal analog "math" artifacts and processes. This may not be a very polite thing to say to a mathematician, but I am wondering if you have impressions along the same line?
It turns out that we all get energy to think and do math and other things from the respiration reaction (organics + oxygen -> water + carbon dioxide +energy). And basically, what that means, if you remember your biology or organic chemistry, is, body-wide, within our cells is a ~steady creative flow of about 10^20 water molecules per second -- coming from the 160 kg of O2 we each respire each year. Generally, each water molecule is sort of tetrahedral in shape with two positive and two negative vertices and so, it turns out that there are at least six ways each water molecule can orient within an enfolding field when it first comes into being at a respiration site. That also means that a chain of n-molecules can form in 6^n different ways. Thus a sequence of 12 molecules could form in 6^12, or about 2 billion different ways. A chain of eighteen molecules could associate with 6^18 or 10^14 different impressions. Now, in this analog math theory, I am assuming that repeating vibrations in the environment ought to result in formation of similar stacks and chains of structurally coded water molecules being formed. This gets us a rather crude image of the vibrations of our internal and external environment forming an internal echo or representation within this active internal analog "math", or "language".
I say it's active because the 6^n stacks of water molecules are really also structurally coded hydrogen-bonding packets and such things, when they unfurl, are connected with and influential in protein-formation and protein-folding, which is to say, memory formation and muscle movement, which is to say, in our case, ALL human expression, perhaps beginning with our nearly universal actions and impressions of counting each of our ten fingers and ten toes, and the like.
Bizarre stuff, huh? Lots of little internal Turin devices writing out structural coded signals.
I'm wondering if mathematicians are taught this type of internal analog math as the basis of the abstract math symbols and expressions, or if they are given different associations or impressions, perhaps leaving it that there is just an uncanny (and unknown) relationship between much or all of nature and math?
Also, I vaguely see the similarity between 2^n binary or boolean math and the 6^n "multiple-state structural coding" that I've made up or stumbled onto. I expect the trend continues with starting with other polyhedra which have limited orientations "within enfolding fields" -- when a containing structure is added. My general hunch is the initial condition IS actually significant for us and we can immediately get to multiple states (relevant to ~quantum mechanics/quantum gravity) by starting with tetrahedron and adding the enfolding cube container, rather than the way it's done presently of beginning with the xyz-cubic framework and adding variants.
Initial conditions do matter in mathematics, don't they?
Before digital computing took over completely, analog computing was dominant for a short while. An analog computer is based on the creation of a model which represents the problem to be solved. But mathematical models of problems can be created of ( almost ) any problem and these models can be implemented on a digital computer. A digital computer is nothing more than a convenient, fast, Turing Machine or equivalent thereof, i.e. the URM or Abacus. And from Mathematical Logic ( Goedel ) we know that these systems have its limitations. It is theoretically impossible to create a program that solves all mathematical problems. - But physicists and biologists say ( and why should we disagree? ) that we -are- computer ( brain ) controlled machines.
Is that a paradox? Humans can do more than computers, we can solve mathematical problems, in fact we -created- the concept of a 'Turing Machine'. This leads us to Roger Penrose. In The Emperor's New Mind, 1999 he claims that artificial intelligence in computers is impossible. He argued that the human brain must exploit a type of physics that he described as 'non-computable'. By this he means beyond algorithmic computing, and thus digital computing.
A picture that keeps fascinating me is that of a predator bird flying high over its prey before, at a carefully -chosen- moment, it makes the dive and following kill. And this is all done with a tiny bird brain. The best comparable thing made by humans thus far is the drone. A huge flying case loaded with bombs operated by a battery of digital computers assisted by human -computers-. Although humans have created a model of a flying bird, it is operated by a human computer on the ground.
Analog computers were special purpose computers, designed to solve one specific problem. A predator bird will never be able to learn new behavior, it cannot be trained to live with chickens. Not immediately anyaway, if ´evolution´ made the bird.
Let me summarize before this turns into a rant.
- There are other models of computing than the Turing machine, i.e. analog computing, brain computing.
- Digital computing is superior over analog computing, brain computing is superior over digital computing.
- Analog and digital computing are human creations we fully understand.
- We don't understand brain computing (yet?).
- Mathematical logic and computability theory study algorithmic ( digital ) computing.
Goedels theorems are somewhat like Russell's paradox in set theory. Goedel's incompleteness theorems are statements about logic and number theory deduced in and with the rules of logic.
Took a day off study today. Have been putting a lot of time in studying lately. Felt like work instead of fun. We have a mini-heatwave in The Netherlands. For the 1st of October it was the hottest day ever ( since recorded weather anyway ). I don't like summers, especially when they turn up in my favorite season autumn. I mean, I think everybody has been off-schedule today. - Tomorrow, I take a day off as well: I really missed working with Mathematica. I got really interested in the foundations of computer science lately. Will read about formal languages, grammars and parsers tomorrow. And of course will have a look at the Mathematica built Lisp interpreter. - I am working on a program myself, SceneGraphica, I need to spend time on that as well. I think I will start all over. Nothing will be lost though. I wouldn't have had the ideas I have now without the effort put in the early versions.
( ... ) mathematics because it is an extremely difficult road to traverse. The terrain is extremely demanding. The amount of work and concentration required to build the foundation necessary to continue extending the framework is immense. ( ... ) - David Andrews
Mathematics, as if you have never seen a skyscraper and are traversing the streets of Manhattan. With that mindset, you can only think that people -walk- to the 60th floor... Anyway, feeling overwhelmed by the sheer size and complexity of maths is not going to help. Only people willing to teach, without ulterior selfish motives, can help. One can write a book about mathematics to impress peers, as a way to meet publication quotas or to -teach-. Like the book The Limits of Mathematics does for example. It is a clear taste of the best mathematics has to offer, an invitation to go on to the next level.
The table of contents says it all:
- Randomness in arithmetic and the decline and fall of reductionism in pure mathematics
- Elegant LISP programs
- An invitation to algorithmic information theory
- The limits of mathematics
- Appendix. LISP interpreter in Mathematica
The appendix contains the source code of a Lisp interpreter coded in Mathematica. I love that. But the book starts with a clear description of the massive changes taking place in ( the thinking about ) mathematics during the first half of the twentieth century, from Hilbert to Turing.
Ramanujan's genius (r) was discovered by Hardy (l) At a very young age Ramanujan designed the following formula for a 3 by 3 magic sq...
Welcome to The Bridge
Mathematics: is it the fabric of MEST?
This is my voyage
My continuous mission
To uncover hidden structures
To create new theorems and proofs
To boldly go where no man has gone before
(Raumpatrouille – Die phantastischen Abenteuer des Raumschiffes Orion, colloquially aka Raumpatrouille Orion was the first German science fiction television series. Its seven episodes were broadcast by ARD beginning September 17, 1966. The series has since acquired cult status in Germany. Broadcast six years before Star Trek first aired in West Germany (in 1972), it became a huge success.)
